The Relations and Functions Class 12 NCERT Solutions for Class 12 Mathematics Chapter 1 Relations and Functions Miscellaneous Exercise present a full step-by-step solutions to every question in this set. Each step in the Relations and Functions Class 12 NCERT Solutions states the rule used and shows the calculation in line, in agreement with the 2026-27 NCERT method.
- CBSE Weightage: 5-7 marks (Relations and Functions + Inverse Trig chapter cluster)
- JEE Main Weightage: 3-5% (1-2 questions on functions / composition / inverse)
- Exercise Profile: 19 mixed questions spanning every concept in Sections 1.2 to 1.5
Student Pulse - Relations and Functions Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
These solutions were prepared by Collegedunia's senior Mathematics faculty against the 2026-27 NCERT print, with every step numbered, every domain-range argument written out, and every inverse verified by composition. The same teachers handle our Class 12 board crash courses, so the working you read here mirrors the marking-scheme style CBSE expects.
Why the Miscellaneous Exercise Matters for Class 12th Boards
The the PDF address this in the same order as the NCERT textbook.
Out of the 19 questions, roughly 12 are direct CBSE-board templates and 4 map straight to JEE Main. The remaining 3 (Q15, Q16, Q19 of NCERT) are objective-type problems that the new NCERT has retained specifically because they teach abstract reasoning.
Students who skip this exercise routinely drop 5 to 7 marks in the board paper because the long-answer question is almost always lifted from this set.
The exercise also doubles as a diagnostic. If you can finish Q1, Q3, Q7, Q9, and Q14 without referring to the NCERT Solutions Class 12 Maths, you have functionally learn the this chapter. If you stall on the equivalence-class proof in Q4 or the binary-operation properties in Q9, that is a signal to revisit Sections 1.2 and 1.5 before moving on.
Relations and Functions Miscellaneous Video Walkthrough
Source: Magnet Brains on YouTube
How the NCERT Solutions Class 12 Maths on the Relations and Functions Class 12 NCERT Solutions Help You
These notes address this in the same order as the NCERT textbook.
Each of the 19 step-by-step solutions in our PDF follows the same four-line structure that CBSE examiners reward:
- Statement of what is to be shown (1 line) - so the examiner sees you understood the demand.
- Property check (reflexive / symmetric / transitive or one-one / onto) - written as a labelled sub-claim.
- Algebraic verification with every substitution shown, no skipped steps.
- Concluding sentence that names the property proved (equivalence relation, bijection, inverse, etc.).
This is the same step grammar our Collegedunia faculty use when they mark sample-paper answer sheets for CBSE pre-board mocks, so practising with these solutions trains the muscle memory needed in the exam hall. Students who follow this four-line structure consistently score 5/6 or 6/6 on the long-answer function question.
Concepts Tested in the Class 12 Maths Chapter 1 Miscellaneous Exercise
The this Class 12 page address this in the same order as the NCERT textbook.
The Miscellaneous set is structured as a concept-mixer. The table below maps each question cluster to the underlying section so you can revise the right theory before attempting the problem.
| Question Cluster | Underlying Concept | NCERT Section | Marks Range (CBSE) |
|---|---|---|---|
| Q1, Q2 | Composition of functions, f ∘ g vs g ∘ f | 1.3 Composition | 3-4 |
| Q3, Q4, Q5 | Equivalence relations and equivalence classes | 1.2 Types of Relations | 5-6 |
| Q6, Q7, Q8 | One-one, onto, invertibility proofs | 1.3, 1.4 Inverse | 4-6 |
| Q9, Q10, Q11 | Binary operations, identity, inverse element | 1.5 Binary Operations | 4-5 |
| Q12 - Q15 | Mixed: bijection + algebraic manipulation | 1.3, 1.4 | 4-6 |
| Q16 - Q19 | Objective / counting / abstract reasoning | All sections | 1-2 (MCQ) |
Notice how heavily Sections 1.3 and 1.4 are weighted. That is your revision priority list: composition and inverse first, equivalence relations second, binary operations third.
Top 5 Formulae and Properties You Need Before Attempting
The 19 questions assume you have these five tools at your fingertips. Memorise them first; the NCERT Solutions Class 12 Maths become 60% shorter.
| # | Tool | Form |
|---|---|---|
| 1 | Reflexive test | (a, a) ∈ R for every a ∈ A |
| 2 | Symmetric test | (a, b) ∈ R ⇒ (b, a) ∈ R |
| 3 | Transitive test | (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R |
| 4 | Inverse rule | f is invertible f is bijective; f-1(y) = x f(x) = y |
| 5 | Composition identity | (g ∘ f)(x) = g(f(x)) ; f ∘ f-1 = IY , f-1 ∘ f = IX |
Full learn formula sheet: Class 12 Maths Chapter 1 Relations and Functions Formula Sheet - 22 formulas with derivations and solved mini-examples.
Question-by-Question Solution Index for the Miscellaneous Exercise
Below is the index of all 19 Miscellaneous Exercise problems with the technique applied. Open the NCERT Solutions Class 12 Maths for the full step-by-step solution to each.
| Q No. | Question Statement (short) | Technique Used |
|---|---|---|
| 1 | Show f: R → x ∈ R : -1 < x < 1 , f(x) = x/(1+|x|) , is one-one onto | Case split on x ≥ 0 and x < 0 , explicit inverse f-1(y) = y/(1-|y|) |
| 2 | Show f: N → N , f(n) = n2 , is one-one but not onto | Direct injection check; counterexample y = 3 shows non-surjective |
| 3 | Number of equivalence relations containing (1,2) on {1,2,3} | Enumerate equivalence classes; answer is 2 |
| 4 | R on Z defined by aRb |a-b| is even, is equivalence | Verify reflexive, symmetric, transitive on parity |
| 5 | Reflexive, symmetric, transitive check for given relation on 1,2,3 | Direct three-property verification |
| 6 | Show f(x) = 4x/(3x+4) on R - -4/3 is invertible | Solve y = 4x/(3x+4) for x; f-1(y) = 4y/(4-3y) |
| 7 | f(n) = n-1 if n odd, n+1 if n even; show invertible on N | Pair-swap argument; f-1 = f |
| 8 | Show f: A → B is bijective for specified A, B and the rule given | One-one by injection, onto by solving for preimage |
| 9 | Binary operation a * b = a + ab on Q ; check identity, inverse | Test associativity, find identity e = 0 , invertibility analysis |
| 10 | Number of binary operations on a set of 2 elements | Counting argument nn2 = 24 = 16 |
| 11 | F = {(a,b) : a, b in A} - check relation properties on specified A | Apply reflexive/symmetric/transitive directly |
| 12 | Prove f ∘ g = g ∘ f = I for given inverse pair | Compute both compositions, simplify |
| 13 | Show specified function is one-one but not onto | Show distinct inputs go to distinct outputs, find element with no preimage |
| 14 | For binary op on R , find identity and inverse | Solve a * e = a for e, then a * b = e for inverse |
| 15 | Find inverse of given bijective function on a real interval | Make x the subject of y = f(x) , swap variables |
| 16 | MCQ on number of reflexive relations on a 3-element set | Counting argument; answer derived from 2n2 - n |
| 17 | MCQ on equivalence relations | Definition-based elimination |
| 18 | MCQ on f: R → R, f(x) = 3x being bijective | One-one and onto via algebra |
| 19 | MCQ on number of one-one functions from 1,2,3 onto itself | Permutation count 3! = 6 |
Roughly 60% of these techniques recur verbatim in CBSE 2021-2025 board papers, which is why the Miscellaneous set has the highest ROI per hour spent in the entire chapter.
Common Mistakes Students Make in the Miscellaneous Exercise
The the resource are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
Previous Year Question Trends - Relations and Functions Chapter
The chapter has been a steady 5 to 7 mark contributor in CBSE Class 12 Maths board papers, with the long-answer slot almost always sourced from the Miscellaneous Exercise pool.
| Year | CBSE Marks | Question Type | Source Pattern |
|---|---|---|---|
| 2025 | 6 | Equivalence relation proof on Z | Modelled on Misc Q4 |
| 2024 | 5 | Bijection + inverse computation | Modelled on Misc Q1, Q6 |
| 2023 | 4 | One-one onto check, no inverse asked | Modelled on Misc Q2, Q13 |
| 2022 | 6 | Equivalence relation + equivalence classes | Modelled on Misc Q3, Q4 |
| 2021 | 5 | Composition + inverse verification | Modelled on Misc Q12, Q15 |
Full year-wise PYQ map: Class 12 Maths Chapter 1 Relations and Functions Notes - includes the full chapter-level PYQ breakdown by topic.
Quick Revision Tips Before You Attempt the Miscellaneous Exercise
Related Resources for Class 12 Maths Chapter 1
- Class 12 Maths Chapter 1 Relations and Functions Notes
- Class 12 Maths Chapter 1 Relations and Functions Formula Sheet
- Class 12 Maths Chapter 1 Relations and Functions NCERT Book PDF
- Class 12 Maths Chapter 1 Relations and Functions Exemplar Solutions
- Class 12 Maths Chapter 1 Relations and Functions Handwritten Notes
NCERT Solutions for Class 12 Maths: All Chapters
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Chapter No. | Chapter Title - NCERT Solutions |
|---|---|
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
NCERT Solutions Class 12 Maths: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
the chapter notes: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Relations and Functions Chapter
The Relations and Functions chapter splits into 2 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 1.1 | Empty, universal, reflexive, symmetric, transitive relations |
| Exercise 1.2 | Injective, surjective, bijective functions; composition |
| Miscellaneous Exercise | Mixed concepts; bijection-invertibility and counting |
PDF Download Formats and Languages for the Relations and Functions Chapter
The Relations and Functions Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The relations and functions class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the relations and functions class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The relations and functions class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The relations and functions class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The relations and functions class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this relations and functions class 12 pdf - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the relations and functions class 12 formulas pdf), and the full relations and functions class 12 pdf on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Relations and Functions Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Relations and Functions chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable relations and functions class 12 important questions you will see on board day.
- relations and functions class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The relations and functions class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching relations and functions class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Relations and Functions chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full relations and functions class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Relations and Functions Notes Pair with NCERT Solutions and the Formula Sheet
The Relations and Functions Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Relations and Functions Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| the PDF PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| relations and functions class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The this chapter cover every back-of-chapter exercise plus the miscellaneous exercise.
- The relations and functions class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The relations and functions class 12 formulas reference sheet is the same A4 file students sometimes refer to as relations and functions class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board relations and functions class 12 textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 relations and functions solutions, class 12 relations and functions ncert solutions, ncert class 12 relations and functions solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to Relations and Functions Class 12 |
|---|---|
| RD Sharma Class 12 Relations and Functions | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Relations and Functions | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo relations and functions class 12 | Free online walkthroughs; useful for video-style learning. |
| Shaalaa relations and functions class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board relations and functions class 12 textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Relations and Functions | Advanced problems for JEE Main/JEE Advanced preparation. |
All NCERT Solutions for Relations and Functions Misc with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 1 Relations and Functions Misc is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
Show that the function f : R → x ∈ R : -1 < x < 1 defined by f(x) = x1 + |x|, x ∈ R, is one one and onto function.
Show that the function f : R → R given by f(x) = x3 is injective.
Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows: For subsets A, B in P(X), A R B if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer.
Find the number of all onto functions from the set 1, 2, 3, …, n to itself.
Let A = -1, 0, 1, 2, B = -4, -2, 0, 2 and f, g : A → B be functions defined by
f(x) = x2 - x, x ∈ A and g(x) = 2|x - 12| - 1, x ∈ A.
Are f and g equal? Justify your answer.
(Hint: One may note that two functions f : A → B and g : A → B such that f(a) = g(a) for all a ∈ A, are called equal functions.)
Let A = 1, 2, 3. Then the number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is:
(A) 1 (B) 2 (C) 3 (D) 4.
Let A = 1, 2, 3. Then the number of equivalence relations containing (1, 2) is:
(A) 1 (B) 2 (C) 3 (D) 4.
How to Use the Relations and Functions Notes Page Most Effectively
The recommended study plan for the Relations and Functions Class 12 chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use these notes PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The relations and functions class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Relations and Functions Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in the Miscellaneous Exercise of Class 12 Maths Chapter 1?
Ans. The Miscellaneous Exercise of Class 12 Maths Chapter 1 Relations and Functions contains 19 questions. Q1 to Q15 are subjective long-answer or short-answer problems and Q16 to Q19 are objective MCQ-type questions.
Ques. Is the Miscellaneous Exercise important for the CBSE Class 12 board exam?
Ans. Yes. The CBSE 5-mark or 6-mark long-answer question on Relations and Functions in the board paper is almost always modelled on a Miscellaneous Exercise problem. Skipping it routinely costs students 5 to 7 marks.
Ques. Which Miscellaneous Exercise questions are most important?
Ans. Q1, Q3, Q4, Q6, Q7, and Q9 are the highest-yield. They cover equivalence-relation proofs, invertibility, and binary operations - the exact three sub-topics CBSE rotates through year after year.
Ques. Are these solutions aligned with the 2026-27 NCERT syllabus?
Ans. Yes. the this chapter is fully aligned with the current 2026-27 NCERT print of Class 12 Maths. Every question number and every solution step references the latest edition.
Ques. Can the same Miscellaneous Exercise questions help with JEE Main?
Ans. Yes. JEE Main consistently uses one to two MCQs per year drawn from the function-composition and invertibility patterns in this exercise. Q1, Q6, and Q9 are particularly close to JEE templates.
Ques. What is the difference between Exercise 1.4 and the Miscellaneous Exercise?
Ans. Exercise 1.4 only tests binary operations in isolation. The Miscellaneous Exercise mixes binary operations with relations, functions, composition, and invertibility - so it tests whether you can recognise which technique applies, not just apply a single named technique.
Ques. Where can I download the Class 12 Maths Chapter 1 Miscellaneous Exercise PDF for free?
Ans. The free PDF of all 19 Miscellaneous Exercise solutions is available at the top of these notes. Click the Download PDF button to save the complete step-by-step step-by-step solutions to your device.
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